How can I find the general formula for the sum of this series?
$$ \sum_{i=0}^n a^ib^{n-i} $$
Where $a$ and $b$ are unrelated constants?
I don't think you can split it into $ \sum_{i=0}^n a^i $ and $ \sum_{i=0}^n b^{n-i} $ right?
Is there a way to simplify this? I'm not sure how to start this. Thanks!
Notice that the sum can be rearranged as
$$b^n \sum_{i = 0}^n \left(\frac a b\right)^i$$
which is now in the form of an explicitly computable geometric series.